1

CAGING OF RIGID POLYTOPES VIA DISPERSION CONTROL OF POINT FINGERS

Peam PipattanasompornAdvisor: Attawith Sudsang

2

Motivation?

!

!

!

3

Better Approach?

4

OverviewMaster Thesis

ProposedPh.d. Thesis

Additional Chapters

XL

H

S C

Fix Cage(2011)

Imperfect Shape (2010)

Robust Cage(2012)

n-Squeeze(2008)

n-Stretch(2008)

2-Squeeze(2006)

2-Stretch(2006)

5

Fix Cage(2011)

n-Stretch(2008)

2-Stretch(2006)

OverviewMaster Thesis

ProposedPh.d. Thesis

Additional Chapters

XL

H

S C Imperfect Shape (2010)

Robust Cage(2012)

n-Squeeze(2008)

2-Squeeze(2006)

6

2-Squeeze, How?

• Keep distance below a value• Given object shape, solve:– Where to place the fingers?– The upperbound distance?

H“Distance”

7

2-Squeeze

• Possible escape path (object frame)

HAlong the path

Distance

8

2-Squeeze

• “Better” escape path

HDistance

“Better”

Upperbound

Initial

Along the path

9

2-Squeeze

• Find an Optimal Escape Path in C-Free

HWorkspace (2D)

b

a

Configuration Space (4D)

(abstracted)C-Obstacle

(a,b)

Abstracted set ofescape configurations

10

2-Squeeze

• Find an Optimal Escape Path in C-Free

Configuration Space (4D)

(abstract)C-Obstacle

Abstract set ofescape configurations

(a,b)

11

C-Free Decomposition

C-Obstacle

12

Paths connecting Terminals

C-Obstacle

13

Finite Categorization of Paths

C-Obstacle

14

Straight Path

Distance : |a-b|2

Along the path

(linear interpolation)

ba

(a,b)

15

Moving Across Convex Subsets

C-Obstacle

16

Through Convex Intersections

C-Obstacle

17

Requirements For The Algorithm

Distance(x)

Rigid Transformation InvariantConvex

&x

18

Convex & RTI Examples

• d1 + d2 + d3

• d12 + d2

2 + d32

• max(d1, d2, d3)

d3 d1

d2

x1

x2x3

• Larger Loose cage• Fingers at a point Smallest

“Formation Size”

19

Results (n-Squeeze)Size: d1

2+d22+d3

2+d42

20

1

23

1

23

1

23

Squeezing?

1

23

1-DOF Scaling ONLY

21

“Size” & “Deformation”

1

23Reference Formation

1

2

3Same sizeNo deformation

Larger sizeDeformed

1

2

3

Smaller sizeSlightly Deformed

1

23

22

Smaller sizeSlightly Deformed

1

23

1

2

3Same sizeNo deformationSame Formation

Larger sizeDeformed

1

2

3

“Size” & “Deformation”

Reference Formation

1

23

23

Smaller sizeSlightly Deformed

1

23

1

2

3Same sizeNo deformationSame Formation

Larger sizeDeformed

1

2

3

“Size” & “Deformation”

Reference Formation

12

3

24

Smaller sizeSlightly Deformed

1

23

1

2

3Same sizeNo deformationSame Formation

Larger sizeDeformed

1

2

3

“Size” & “Deformation”

Reference Formation

12

3

25

“Size” & “Deformation”

• |r|22(x) = |A†x|2

2

– “Scale” or “Size” (w.r.t. reference)

• D(x) = |A(r; t) – x|22

– “Deformation upto Scale” (w.r.t. reference)

1

2

3

12

3

A stores information of the reference.

26

Squeezing ?

1

23

1

23

1

23

1-DOF Scaling ONLY

Size = |r|22 < ???

D ≤ 0&

Convex& RTI

Convex& RTI

27

Squeezing

1

23

1

23

1

23

1-DOF Scaling ONLY

Size = |r|22 < ???

D ≤ 0&

x

D > 0D > 0

D ≤ 0

|r|22 ; D ≤ 0

; D > 0Size* =

28

Fix Formation Cage

1

23

Size* = 1

ConvexConstraint

Convex& RTI

Size* 1Size* ≤ 1

“Stretch”“Squeeze” &

29

Robust Caging

Independent Capture Regions

• Keep error (deformation) below a value

• Given object shape, find:– Where to place the fingers– The upperbound error

30

n-Squeeze vs Fix Formation

KEEP SIZE ERROR (DEFORMATION)BELOW UPPERBOUND BELOW UPPERBOUND

OPTIMAL ESCAPE PATH SIZE MINIMIZEUPPERBOUND DISTANCE ERROR (DEFORMATION)

X

31

Error Tolerance

inf r,tϵR2

223 3

1 1

2

23

1

D2 =23

1

inf |r|2=1tϵR2

223 3

1 1

p

23

1

Ep =23

1

NOT CONVEX!

r + t -

r + t -

“Placement Error”

“Placement Error upto Scale”

32

Approximation

inf g(r)|r|2=1

inf g(r)r ϵ Ri

mini ϵ{1,…, m}

R1

R2

R3

R4

33

Approximation

inf g(r)|r|2=1

inf g(r)r ϵ Ri

mini ϵ{1,…, m}

R1

R2

R3

R4

Min of Convex Functions

(not convex)

Optimal Path

Min of a Convex Function is Convex

f = f1 = min(f1)

35

Optimal Path

Min of Two Convex Functions

f = min(f1, f2)

f1 = f f = f2f1 = f = f2

35

36

Optimal Path

Min of Two Convex Functions

f = min(f1, f2)

f1 = f f = f2f1 = f = f2

36

???

Optimal Path

21

37

x

f(x)f1=f=f2

f=f2f=f1

What is the optimal path, starting from the minimal points?

Critical Point

21

38

x

f1=f=f2

f=f2f=f1

Consider…

1,2

Only the points under the water level are reachable when the maximum deformation is limited to below the water level.

f(x)

Optimal Path

1,2

21

39

x

f1=f=f2

Critical Value

f=f2f=f1

: minimizer for a CONVEX optimization problem:minimize L s.t.f1(x) < Lf2(x) < L f(x)

Critical Point

1,2

21

40

x

f1=f=f2

Critical Value

f=f2f=f1

f(x)

41

Min of Multiple Convex Functions

Min of Multiple Convex Functions

f = min(f1, f2 , f3)

f= f1

f= f2

f= f3

42

Min of Multiple Convex Functions

Min of Multiple Convex Functions

f = min(f1, f2 , f3)

f= f1

f= f2

f= f3

1,2

2,3

1,31

32

43

Search Space

Min of Multiple Convex Functions

f = min(f1, f2 , f3)

1,2

2

1

3

1,3

2,3

Include all possible between any two regions: f=fi , f=fj

44

Optimal Path

45

Results

46

Results

47

Shape Uncertainty

Exact Object(Unknown) Scanned Objectsensor

48

Idea

• Cage subobject Cage object ?• Fingers must not penetrate the object.

H

49

Exact Object(Unknown)

Idea

Exact boundary (unknown) but inbetween the bounds.

• Find placements that cage subobject, outside superobject.

50

Applications

• Simplification• Curved Surface, Spherical Fingers• Shape Uncertainty• Slightly Deformable Object• Partial Observation

51

Results

52

Results

53

Conclusion

• O(c log c) exact algorithms– Squeeze, Stretch, Squeeze & Stretch– c : # decomposed convex features

• O(cm2 log( cm2 ) ) approximate algorithm– m : # approximation facets

• Extension to three dimension.• Trade error tolerance with uncertainty.

54

Q&A

:V :o

55

Publications• Journal Papers

– Peam Pipattanasomporn, Attawith Sudsang: Two-Finger Caging of Nonconvex Polytopes. IEEE Transactions on Robotics 27 (2011)

– Thanathorn Phoka, Pawin Vongmasa, Chaichana Nilwatchararang, Peam Pipattanasomporn and Attawith Sudsang: Optimal independent contact regions for two-fingered grasping of polygon. Robotica (2011)

• Conference Papers– Peam Pipattanasomporn, Attawith Sudsang: Object caging under imperfect shape knowledge. ICRA 2010 – Thanathorn Phoka, Pawin Vongmasa, Chichana Nilwatchararang, Peam Pipattanasomporn, Attawith

Sudsang: Planning optimal independent contact regions for two-fingered force-closure grasp of a polygon. ICRA 2008

– Peam Pipattanasomporn, Pawin Vongmasa, Attawith Sudsang: Caging rigid polytopes via finger dispersion control. ICRA 2008

– Peam Pipattanasomporn, Pawin Vongmasa, Attawith Sudsang: Two-Finger Squeezing Caging of Polygonal and Polyhedral Object. ICRA 2007

– Peam Pipattanasomporn, Attawith Sudsang: Two-finger Caging of Concave Polygon. ICRA 2006 – Thanathorn Phoka, Peam Pipattanasomporn, Nattee Niparnan, Attawith Sudsang: Regrasp Planning of

Four-Fingered Hand for Parallel Grasp of a Polygonal Object. ICRA 2005